S. Huet
A. Bouvier
M.-A. Poursat
E. Jolivet
Statistical Tools for
Nonlinear Regression
A Practical Guide With S-PLUS and
R Examples
Second Edition
Contents
Preface t o the Second Edition XI Preface t o t h e First Edition XIII 1 Nonlinear Regression Model and Parameter Estimation . . . 1
1.1 Examples 1 1.1.1 Pasture Regrowth: Estimating a Growth Curve 1
1.1.2 Radioimmunological Assay of Cortisol: Estimating a
Calibration Curve 2 1.1.3 Antibodies Anticoronavirus Assayed by an ELISA
Test: Comparing Several Response Curves 6 1.1.4 Comparison of Immature and Mature Goat Ovocytes:
Comparing Parameters 8 1.1.5 Isomerization: More than One Independent Variable . . . 9
1.2 The Parametric Nonlinear Regression Model 10
1.3 Estimation 11 1.4 Applications 13
1.4.1 Pasture Regrowth: Parameter Estimation and Graph
of Observed and Adjusted Response Values 13 1.4.2 Cortisol Assay: Parameter Estimation and Graph of
Observed and Adjusted Response Values 13 1.4.3 ELISA Test: Parameter Estimation and Graph of
Observed and Adjusted Curves for May and June 14 1.4.4 Ovocytes: Parameter Estimation and Graph of
Observed and Adjusted Volume of Mature and
Immature Ovocytes in Propane-Diol 15 1.4.5 Isomerization: Parameter Estimation and Graph of
Adjusted versus Observed Values 16
Contents
Accuracy of Estimators, Confidence Intervals and T e s t s . . . . 29
2.1 Examples 29 2.2 Problem Formulation 30
2.3 Solutions 30 2.3.1 Classical Asymptotic Results 30
2.3-2 Asymptotic Confidence Intervals for A 32 2.3.3 Asymptotic Tests of A = Ao against A ^ Ao 33 2.3.4 Asymptotic Tests of A6 = Lo against AO ^ L0 34
2.3.5 Bootstrap Estimations 35
2.4 Applications 38 2.4.1 Pasture Regrowth: Calculation of a Confidence
Interval for the Maximum Yield 38 2.4.2 Cortisol Assay: Estimation of the Accuracy of the
Estimated Dose D 39 2.4.3 ELISA Test: Comparison of Curves 40
2.4.4 Ovocytes: Calculation of Confidence Regions 42 2.4.5 Isomerization: An Awkward Example 43 2.4.6 Pasture Regrowth: Calculation of a Confidence
Interval for A = exp 63 47
2.5 Conclusion 49 2.6 Using nls2 49
Variance Estimation 61
3.1 Examples 61 3.1.1 Growth of Winter Wheat Tillers: Few Replications . . . . 61
3.1.2 Solubility of Peptides in Trichloacetic Acid Solutions:
No Replications 63 3.2 Parametric Modeling of the Variance 65
3.3 Estimation 66 3.3.1 Maximum Likelihood Estimation 66
3.3.2 Quasi-Likelihood Estimation 67 3.3.3 Three-Step Estimation 69 3.4 Tests and Confidence Regions 69
3.4.1 The Wald Test 69 3.4.2 The Likelihood Ratio Test 70
3.4.3 Bootstrap Estimations 71 3.4.4 Links Between Testing Procedures and Confidence
Region Computations 72 3.4.5 Confidence Regions 73
3.5 Applications 74 3.5.1 Growth of Winter Wheat Tillers 74
3.5.2 Solubility of Peptides in Trichloacetic Acid Solutions... 78
Contents VII
Diagnostics of Model Misspecification 93
4.1 Problem Formulation 93 4.2 Diagnostics of Model Misspecincations with Graphics 94
4.2.1 Pasture Regrowth Example: Estimation Using a
Concave-Shaped Curve and Plot for Diagnostics 95 4.2.2 Isomerization Example: Graphics for Diagnostic 95 4.2.3 Peptides Example: Graphics for Diagnostic 97 4.2.4 Cortisol Assay Example: How to Choose the Variance
Function Using Replications 99 4.2.5 Trajectory of Roots of Maize: How to Detect
Correlations in Errors 103 4.2.6 What Can We Say About the Experimental Design? . . . 107
4.3 Diagnostics of Model Misspecincations with Tests 110 4.3.1 RIA of Cortisol: Comparison of Nested Models 110
4.3.2 Tests Using Replications 110 4.3.3 Cortisol Assay Example: Misspecification Tests Using
Replications 112 4.3.4 Ovocytes Example: Graphics of Residuais and
Misspecification Tests Using Replications 112 4.4 Numerical Troubles During the Estimation Process: Peptides
Example 114 4.5 Peptides Example: Concluded 118
4.6 Using nls2 119
Calibration and Prediction 135
5.1 Examples 135 5.2 Problem Formulation 137
5.3 Confidence Intervals 137 5.3.1 Prediction of a Response 137
5.3.2 Calibration with Constant Variances 139 5.3.3 Calibration with Nonconstant Variances 141
5.4 Applications 142 5.4.1 Pasture Regrowth Example: Prediction of the Yield at
Time x0 = 50 142
5.4.2 Cortisol Assay Example 143 5.4.3 Nasturtium Assay Example i 144
5.5 References 145 5.6 Using nls2 145
Binomial Nonlinear Models 153
6.1 Examples 153 6.1.1 Assay of an Insecticide with a Synergist: A Binomial
Nonlinear Model 153 6.1.2 Vaso-Constriction in the Skin of the Digits: The Case
VIII Contents
6.1.3 Mortality of Confused Flour Beetles: The Choice of a
Link Function in a Binomial Linear Model 156 6.1.4 Mortality of Confused Flour Beetles 2: Survival
Analysis Using a Binomial Nonlinear Model 158 6.1.5 Germination of Orobranche: Overdispersion 159
6.2 The Parametric Binomial Nonlinear Model 160
6.3 Overdispersion, Underdispersion 161
6.4 Estimation 162 6.4.1 Case of Binomial Nonlinear Models 162
6.4.2 Case of Overdispersion or Underdispersion 164
6.5 Tests and Confidence Regions 165
6.6 Applications 167 6.6.1 Assay of an Insecticide with a Synergist: Estimating
the Parameters 167 6.6.2 Vaso-Constriction in the Skin of the Digits: Estimation
and Test of Nested Models 171 6.6.3 Mortality of Confused Flour Beetles: Estimating the
Link Function and Calculating Confidence Intervals
for the LD90 172 6.6.4 Mortality of Confused Flour Beetles 2: Comparison of
Curves and Confidence Intervals for the ED50 174 6.6.5 Germination of Orobranche: Estimating
Overdispersion Using the Quasi-Likelihood
Estimation Method 177
6.7 Using nls2 180
7 Multinomial and Poisson Nonlinear Models 199
7.1 Multinomial Model 199 7.1.1 Pneumoconiosis among Coal Miners: An Example of
Multicategory Response Data 200 7.1.2 A Cheese Tasting Experiment 200 7.1.3 The Parametric Multinomial Model 201 7.1.4 Estimation in the Multinomial Model 204 7.1.5 Tests and Confidence Intervals 206 7.1.6 Pneumoconiosis among Coal Miners: The Multinomial
Logit Model 208 7.1.7 Cheese Tasting Example: Model Based on Cumulative
Probabilities 210 7.1.8 Using nls2 213 7.2 Poisson Model 221
7.2.1 The Parametric Poisson Model 222 7.2.2 Estimation in the Poisson Model 222 7.2.3 Cortisol Assay Example: The Poisson Nonlinear Model. 223
Contents IX
References 227
I n d e x 231
